Low-lying zeros seminar
- chair: Kaneenika Sinha
- times: Wednesdays 1.30 - 2.30 pm. Please note: The first meeting on 19 Jan will be from 2-3 pm in the Baker boardroom.
Wed. 19 Jan in the Common Room: Henryk Iwaniec, Introduction to low lying zeros of L-functions
- Abstract: The content of the paper "Low lying zeros of families of L-functions" by Iwaniec, Luo and Sarnak will be described in general terms.
- Wed. 26 Jan in Baker boardroom: Henryk Iwaniec, "The paper of Iwaniec, Luo and Sarnak"
- Abstract: We will continue to describe the contents of the above mentioned paper.
- Wed. 9 Feb in Baker boardroom. H. Iwaniec will continue to present the contents of the above paper.
- 16 Feb in Baker boardroom:
- Speaker: David Farmer Title: What you need to know about random matrix theory
- 23 Feb in Baker boardroom, Andrew Knightly, Title: Petersson's fomula and related topics,
- Abstract: A crucial ingredient in the ILS paper is an estimate for a sum of Hecke eigenvalues. This is handled using a variant of the trace formula due to Petersson, but there are some obstacles, especially the fact that Petersson's formula includes the contribution of old forms. I will briefly outline the ILS strategy for isolating the newform contribution when N is square-free. I will then propose an alternate strategy coming from representation theory. Along the way, I will give a very basic introduction to the trace formula, and explain how one can use it to derive various identities of interest in classical analytic number theory. These include:
- The trace of a Hecke operator; The Petersson and Kuznetsov formulas; Formulas for averages/moments of L-functions; Vertical Sato-Tate laws for Hecke eigenvalues.
- Abstract: A crucial ingredient in the ILS paper is an estimate for a sum of Hecke eigenvalues. This is handled using a variant of the trace formula due to Petersson, but there are some obstacles, especially the fact that Petersson's formula includes the contribution of old forms. I will briefly outline the ILS strategy for isolating the newform contribution when N is square-free. I will then propose an alternate strategy coming from representation theory. Along the way, I will give a very basic introduction to the trace formula, and explain how one can use it to derive various identities of interest in classical analytic number theory. These include:
- H. Iwaniec, W. Luo, and, P. Sarnak, Low lying zeros of families of L-functions, Publ. IHES, 2000.
there is also a study guide and reading list by Steven Miller, at Williams: http://www.williams.edu/go/math/sjmiller/public_html/ntandrmt/